Third law of thermodynamics

Thermodynamics
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Introduction

The third law of thermodynamics is sometimes stated as follows:

The entropy of a perfect crystal at absolute zero is exactly equal to zero.

At zero kelvin the system must be in a state with the minimum possible energy, and this statement of the third law holds true if the perfect crystal has only one minimum energy state. Entropy is related to the number of possible microstates, and with only one microstate available at zero kelvin, the entropy is exactly zero.^[1]

A more general form of the third law applies to systems such as glasses that may have more than one minimum energy state:

The entropy of a system approaches a constant value as the temperature approaches zero.

The constant value (not necessarily zero) is called the residual entropy of the system.^[2]

History

The third law was developed by the chemist Walther Nernst during the years 1906-1912, and is therefore often referred to as Nernst's theorem or Nernst's postulate. The third law of thermodynamics states that the entropy of a system at absolute zero is a well-defined constant. This is because a system at zero temperature exists in its ground state, so that its entropy is determined only by the degeneracy of the ground state. It means that "it is impossible by any procedure, no matter how idealised, to reduce any system to the absolute zero of temperature in a finite number of operations".

An alternative version of the third law of thermodynamics as stated by Gilbert N. Lewis and Merle Randall in 1923:

If the entropy of each element in some (perfect) crystalline state be taken as zero at the absolute zero of temperature, every substance has a finite positive entropy; but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances.

This version states not only ΔS will reach zero at 0 K, but S itself will also reach zero as long as the crystal has a ground state with only one configuration. Some crystals form defects which causes a residual entropy. This residual entropy disappears when the kinetic barriers to transitioning to one ground state are overcome. ^[3]

With the development of statistical mechanics, the third law of thermodynamics (like the other laws) changed from a fundamental law (justified by experiments) to a derived law (derived from even more basic laws). The basic law from which it is primarily derived is the statistical-mechanics definition of entropy for a large system:

where S is entropy, k_B is the Boltzmann constant, and is the number of microstates consistent with the macroscopic configuration.

Explanation

In simple terms, the third law states that the entropy of a perfect crystal approaches zero as the absolute temperature approaches zero. This law provides an absolute reference point for the determination of entropy. The entropy determined relative to this point is the absolute entropy. Mathematically, the absolute entropy of any system at zero temperature is the natural log of the number of ground states times Boltzmann's constant k_B.

The entropy of a perfect crystal lattice as defined by Nernst's theorem is zero provided that its ground state is unique, because ln(1) = 0.

An example of a system which does not have a unique ground state is one containing half-integer spins, for which time-reversal symmetry gives two degenerate ground states. For such systems, the entropy at zero temperature is at least ln(2)k_B (which is negligible on a macroscopic scale). Some crystalline systems exhibit geometrical frustration, where the structure of the crystal lattice prevents the emergence of a unique ground state. Ground-state helium (unless under pressure) remains liquid.

In addition, glasses and solid solutions retain large entropy at 0K, because they are large collections of nearly degenerate states, in which they become trapped out of equilibrium. Another example of a solid with many nearly-degenerate ground states, trapped out of equilibrium, is ice Ih, which has "proton disorder".

For the entropy at absolute zero to be zero, the magnetic moments of a perfectly ordered crystal must themselves be perfectly ordered; indeed, from an entropic perspective, this can be considered to be part of the definition of "perfect crystal". Only ferromagnetic, antiferromagnetic, and diamagnetic materials can satisfy this condition. Materials that remain paramagnetic at 0K, by contrast, may have many nearly-degenerate ground states (for example, in a spin glass), or may retain dynamic disorder (a spin liquid ^[4]).

Mathematical formulation

Consider a closed system in internal equilibrium. As the system is in equilibrium there are no irreversible processes so the entropy production is zero. During the heat supply temperature gradients are generated in the material, but the associated entropy production can be kept low enough if the heat is supplied slowly. The increase in entropy due the added heat δQ is then given by

(1)

The temperature rise δT due to the heat δQ is determined by the heat capacity C(T,X) according to

(2)

The parameter X is a symbolic notation for all parameters (such as pressure, magnetic field, liquid/solid fraction, etc.) which are kept constant during the heat supply. E.g. if the volume is constant we get the heat capacity at constant volume C_V. In the case of a phase transition from liquid to solid, or from gas to liquid the parameter X can be the fraction of one of the two components. Combining relations (1) and (2) gives

(3)

Integration of Eq.(3) from a reference temperature T₀ to an arbitrary temperature T gives the entropy at temperature T

(4)

We now come to the mathematical formulation of the third law. There are three steps:

1: in the limit T₀→0 the integral in Eq.(4) is finite. So that we may take T₀=0 and write

(5)

2. the value of S(0,X) is independent of X. In mathematical form

(6)

So Eq.(5) can be further simplified to

(7)

Equation (6) can also be formulated as

(8)

In words: at absolute zero all isothermal processes are isentropic. Eq.(8) is the mathematical formulation of the third law.

3: as one is free to chose the zero of the entropy it is convenient to take

(9)

so that Eq.(7) reduces to the final form

(10)

The physical meaning of Eq.(9) is deeper than just a convenient selection of the zero of the entropy. It is due to the perfect order at zero kelvin as explained before.

Consequences of the third law

Fig.1 Left side: Absolute zero can be reached in a finite number of steps if S(0,X₁)≠S(0, X₂). Right: An infinite number of steps is needed since S(0,X₁)= S(0,X₂).

Can absolute zero be obtained?

The reason that T=0 cannot be reached according to the third law is explained as follows: Suppose that the temperature of a substance can be reduced in an isentropic process by changing the parameter X from X₂ to X₁. One can think of a multistage nuclear demagnetization setup where a magnetic field is switched on and off in a controlled way. ^[5] If there would be an entropy difference at absolute zero T=0 could be reached in a finite number of steps. However, at T=0 there is no entropy difference so an infinite number of steps would be needed. The process is illustrated in Fig.1.

Specific heat

Suppose that the heat capacity of a sample in the LT region can be approximated by C(T,X)=C₀T^α, then

(11)

The integral is finite for T₀→0 if α>0. So the heat capacity of all substances must go to zero at absolute zero

(12)

The molar specific heat at constant volume of a monatomic classical ideal gas, such as helium at room temperature, is given by C_V=(3/2)R with R the molar ideal gas constant. Substitution in Eq.(4) gives

(13)

In the limit T₀→0 this expression diverges. Clearly a constant heat capacity does not satisfy Eq.(12). This means that a gas with a constant heat capacity all the way to absolute zero violates the third law of thermodynamics.

The conflict is solved as follows: At a certain temperature the quantum nature of matter starts to dominate the behavior. Fermi particles follow Fermi-Dirac statistics and Bose particles follow Bose-Einstein statistics. In both cases the heat capacity at low temperatures is no longer temperature independent, even for ideal gases. For Fermi gases

(14)

with the Fermi temperature T_F given by

(15)

Here N_A is Avogadro's number, V_m the molar volume, and M the molar mass.

For Bose gases

(16)

with T_B given by

(17)

The specific heats given by Eq.(14) and (16) both satisfy Eq.(12).

Vapor pressure

The only liquids near absolute zero are ³He and ⁴He. Their heat of evaporation has a limiting value given by

(18)

with L₀ and C_p constant. If we consider a container, partly filled with liquid and partly gas, the entropy of the liquid-gas mixture is

(19)

where S_l(T) is the entropy of the liquid and x is the gas fraction. Clearly the entropy change during the liquid-gas transition (x from 0 to 1) diverges in the limit of T→0. This violates Eq.(8). Nature solves this paradox as follows: at temperatures below about 50 mK the vapor pressure is so low that the gas density is lower than the best vacuum in the universe. In other words: below 50 mK there is simply no gas above the liquid.

Latent heat of melting

The melting curves of ³He and ⁴He both extend down to absolute zero at finite pressure. At the melting pressure liquid and solid are in equilibrium. The third law demands that the entropies of the solid and liquid are equal at T=0. As a result the latent heat of melting is zero and the slope of the melting curve extrapolates to zero as a result of the Clausius-Clapeyron equation.

Thermal expansion coefficient

The thermal expansion coefficient is defined as

(20)

With the Maxwell relation

(21)

and Eq.(8) with X=p it is shown that

(22)

So the thermal expansion coefficient of all materials must go to zero at zero kelvin.

See also

• Adiabatic process
• Ground state
• Laws of thermodynamics
• Residual entropy
• Thermodynamic entropy
• Timeline of thermodynamics, statistical mechanics, and random processes

References

Further reading

• Goldstein, Martin & Inge F. (1993) The Refrigerator and the Universe. Cambridge MA: Harvard University Press. ISBN 0-674-75324-0. Chpt. 14 is a nontechnical discussion of the Third Law, one including the requisite elementary quantum mechanics.